For classical random walks \((X_n)_{n=0}^{\infty }\)  on a space, the walk dimension (or fractal dimension of the walk, diffusion exponent), \(d_w\) , is defined by the reciprocal of the exponent of the expected distance at time \(n\)  from the initial point \(X_0\) ; namely, \(E|X_n-X_0| \sim n^{1/d_{w}}\) , as \(n \rightarrow \infty \) . For simple random walk on \(d\) -dimensional lattice \(Z^d\) , it is well known that \(d_w= 2\) , for all \(d\) . A random walk is called a diffusion if \(d_w =2\) . When \(d_w >2\) , the walk is called an anomalous diffusion. This type of anomalous diffusion were first observed by random walks in random media such as critical percolation clusters. Critical percolation clusters behave like a fractal in a large scale and simple random walks on the clusters are expected to be anomalous diffusions [1]}, [2]}.
